Open Problems on Central Simple Algebras — Problem 6

Definitions and mathematical statement

For a positive integer $n$, let $C_n$ denote a cyclic group of order $n$. Given an action $\alpha\colon C_m\to\operatorname{Aut}(C_p)$, write $C_p\rtimes_{\alpha}C_m$ for the corresponding semidirect product; $C_p\times C_m$ denotes the direct product. The condition that $m$ is prime to $p$ means $\gcd(m,p)=1$.

The supplied source asks for groups $G,G'$ and conditions on a field $F$ under which \[ G\Rightarrow_F G'. \] In particular, it asks whether, for $m$ prime to $p$, \[ C_p\rtimes C_m\Rightarrow_F C_p\times C_m, \] and whether every quasi-symbol is cyclic.

The supplied record does not define the relation $\Rightarrow_F$, the class of quasi-symbols, or the meaning of cyclicity for those objects. Consequently, the objects and quantifiers underlying the relation and the last universal assertion cannot be made explicit from this record. The semidirect-product action is also unspecified, and the record does not explicitly state whether $p$ must be prime. No action, characteristic restriction, root-of-unity hypothesis, or interpretation of the missing terminology is imposed in this draft. These qualifications must be resolved from the source context before the requests constitute fully specified mathematical statements.

Short English statement

Determine when the source's relation $G\Rightarrow_F G'$ holds, especially from $C_p\rtimes C_m$ to $C_p\times C_m$ when $\gcd(m,p)=1$, and determine whether every quasi-symbol is cyclic. The relation, quasi-symbols, and relevant field and action hypotheses are not defined in the supplied record.

Sources

• Local UnsolvedMath registry, permanent ID $800696$, exact source problem code catalogue entry, titled Open Problems on Central Simple Algebras --- Problem 6. The registry attributes the problem to Auel, Brussel, Garibaldi, and Vishne, with proposed year 2010, and supplies the status label open; that label is not an independently verified mathematical or literature assessment.
• Auel, Brussel, Garibaldi, and Vishne, Open Problems on Central Simple Algebras (2010), Problem 6. Supplied source URL: https://arxiv.org/abs/1006.3304. The record reports access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance details; no independent source visit or verification is claimed.